Stability for the electromagnetic inverse source problem in inhomogeneous media

A Proof of Proposition 2.3

Define

By the spectral theorem, we have the following mapping properties for 𝐄 ⁢ ( t ) :

It can be verified that 𝐄 ⁢ ( t ) solves the equations

where □ := ∂ t 2 + 𝒫 . Set □ 0 = ∂ t 2 + ∇ × ( ∇ × ⋅ ) . We use the notation [ A , B ] = A ⁢ B - B ⁢ A for the commutator.

Choose a large enough such that supp ⁡ χ ⊂ B a , and take χ a ∈ C c ∞ ⁢ ( B a ) , χ a ≡ 1 , on supp ⁡ χ . Assume that ψ a ∈ C ∞ ⁢ ( ℝ t × ℝ 3 ) satisfies

and

where ψ a 0 ∈ C c ∞ ⁢ ( T a + R , T a + R + 1 ) is a function of t only.

First, we show that

In fact, let χ 0 , χ 1 ∈ C 0 ∞ ⁢ ( B a ) such that χ 0 ≡ 1 on supp ⁡ χ a and χ 1 ≡ 1 on supp ⁡ χ 0 . The non-trapping condition in Assumption 2.2 implies

Let □ 0 := ∂ t ⁢ t - ∇ × ( ∇ × ⋅ ) . We have

It is clear that ( 1 - χ 0 ) ⁢ 𝐄 ⁢ ( t ) ⁢ χ a satisfies the initial conditions

By the Duhamel principle and the support property of the propagator 𝐄 0 of free-space electromagnetic wave equations in odd dimensions

we have

Since ψ ≡ 0 on { t < T a + R } , relations (A.2) and (A.3) imply (A.1).

Take a function ζ a ∈ C ∞ such that

It follows from the above claim that

Notice that ζ a ⁢ ( x , t ) ≡ 1 for t ≤ 0 . Thus, [ □ , ζ a ] = 0 on { t ≤ 0 } , and consequently 𝐅 a ⁢ ( t ) ≡ 0 for t ≤ 0 .

Choose χ b ∈ C c ∞ ⁢ ( B a ) such that χ b = 1 near B R and χ a ≡ 1 on supp ⁡ χ b . Let 𝐖 a ⁢ ( t ) solve

It is clear that 𝐖 a ⁢ ( t ) ⁢ g ∈ C ∞ ⁢ ( ℝ t ; 𝒟 1 / 2 ) .

In what follows, we show that for g ∈ ℋ ,

Here C a = 2 ⁢ a + T a + 1 . We write

where H ⁢ ( t ) is the Heaviside function. It is easy to see that the first term has a support in

The last term 𝐐 a ⁢ ( t ) solves

Observe that

vanishes on { t ≥ a + T a + 1 } . Therefore, it follows from the sharp Huygen principle that

Notice also that ( 1 - χ b ) ⁢ 𝐅 a ⁢ ( t ) ⁢ g = 0 on { t ≤ | x | + T a } . By the finite speed of propagation, we have

Choose χ c ∈ C c ∞ ⁢ ( B a ) such that χ c = 1 near B R and χ b = 1 on supp ⁡ χ c . Let

A straightforward calculation yields

where

We prove (2.1) by establishing the following estimates:

for any N. Then we have

The invertibility of ℐ + 𝒦 a ⁢ ( λ ) for λ ∈ Ω M is guaranteed by estimate (A.6).

First, we prove (A.5). Since 𝒫 is positive, we have

Thus,

and

Consequently, we have

In view of (A.4), we have

for any N.

Next, we prove (A.6). We only need to notice that χ b ⁢ 𝐅 a ⁢ ( t ) and [ ∇ × ( ∇ × ⋅ ) , χ c ] 𝐖 a ( t ) both vanish outside a compact set in t. Hence, (2.1) is proved for α = 0 .

Now, we prove (2.1) for α = 1 2 . It suffices to prove

and use relation (A.7). Since

we have

Similarly, we obtain

By

and relation (A.4), we get

Then (A.8) is proved.

For α = 1 , we take χ 1 ∈ C c ∞ ⁢ ( ℝ 3 ) satisfying χ 1 = 1 near supp ⁡ χ . A simple calculations gives

which shows (2.1) for α = 1 .